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• 4v-sexprs
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4v-sexpr-<=

Extension of the four-valued lattice ordering to sexprs.

We say X <= Y for sexprs if X always evaluates to a smaller value than Y evaluates to in every environment, in the sense of 4v-<=.

When X <= Y, we sometimes call X a conservative approximation of Y.

Definitions and Theorems

Theorem: 4v-sexpr-<=-necc

(defthm 4v-sexpr-<=-necc
  (implies (not (4v-<= (4v-sexpr-eval x env)
                       (4v-sexpr-eval y env)))
           (not (4v-sexpr-<= x y))))

Theorem: 4v-sexpr-<=-witnessing-witness-rule-correct

(defthm 4v-sexpr-<=-witnessing-witness-rule-correct
  (implies (not ((lambda (env y x)
                   (not (4v-<=$inline (4v-sexpr-eval x env)
                                      (4v-sexpr-eval y env))))
                 (4v-sexpr-<=-witness x y)
                 y x))
           (4v-sexpr-<= x y))
  :rule-classes nil)

Theorem: 4v-sexpr-<=-instancing-instance-rule-correct

(defthm 4v-sexpr-<=-instancing-instance-rule-correct
  (implies (not (4v-<=$inline (4v-sexpr-eval x env)
                              (4v-sexpr-eval y env)))
           (not (4v-sexpr-<= x y)))
  :rule-classes nil)

Theorem: 4v-sexpr-<=-nil

(defthm 4v-sexpr-<=-nil
  (4v-sexpr-<= nil x))

Theorem: 4v-sexpr-<=-refl

(defthm 4v-sexpr-<=-refl
  (4v-sexpr-<= x x))

Theorem: 4v-sexpr-<=-trans1

(defthm 4v-sexpr-<=-trans1
  (implies (and (4v-sexpr-<= a b)
                (4v-sexpr-<= b c))
           (4v-sexpr-<= a c)))

Theorem: 4v-sexpr-<=-trans2

(defthm 4v-sexpr-<=-trans2
  (implies (and (4v-sexpr-<= b c)
                (4v-sexpr-<= a b))
           (4v-sexpr-<= a c)))

Subtopics

4v-sexpr-alist-<=

Extension of 4v-sexpr-<= to alists.